Concentrating solutions for a class of indefinite Schrödinger-Poisson systems with doubly critical growth
DOI:
https://doi.org/10.12775/TMNA.2024.055Keywords
Schrödinger-Poisson system, critical nonlocal term, indefinite variational problem, linking, modified Pankov-Nehari manifold, concentratingAbstract
We study the following Schrödinger-Poisson system involving critical nonlocal term with indefinite steep potential well $$ \begin{cases} -\Delta u+(\lambda V(x)-\mu)u-\phi |u|^3u= f(u), & x\in\R^3, \\ -\Delta \phi= |u|^5, & x\in\R^3, \end{cases} $$% where $\lambda> 0$ is a parameter, $V\in \mathcal{C}\big(\R^3,\R^+\big)$ admits a potential well $\Omega \triangleq\text{int}V^{-1}(0)$, and $\mu> \mu_1$ is a constant such that the operator $L_\lambda \triangleq -\Delta + \lambda V-\mu$ is non-degenerate when $\lambda$ is large enough with $\{\mu_j\}_{j=1}^\infty$ denoting the Dirichlet eigenvalues of $\big(-\Delta,H_0^1(\Omega)\big)$. If $f$ satisfies some suitable assumptions involving critical growth, with the help of a linking-type result involving the modified Pankov-Nehari manifold procedure, we establish the existence and concentrating behavior of positive solutions for the given system using variational methods.References
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